2020/11/09 by Steinerberger, Stefan
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2011.04630
We revisit a classical question: how large is the minimal logarithmic energy of n points on \mathbbS2 Elog(n) = min_x1, …, xn ∈ \mathbbS2 ∑i,j =1 \atop i ≠ jn log(1)/(‖xi-xj‖) ? Betermin & Sandier (building on work of Sandier & Serfaty) showed that Elog(n) = ( (1)/(2) - log2 )n2 - \fracn logn2 + clog ⋅ n + o(n), where the constant clog is characterized by a certain renormalized minimization problem. Brauchart, Hardin & Saff conjectured a closed form expression for clog (∼ -0.05) assuming analytic continuation. We describe a simple renormalization approach that results in a purely local problem involving superpositions of Gaussians. In particular, if the hexagonal lattice minimizes Gaussians energy, this would prove that clog indeed coincides with the conjectured value. We also improve the lower bound from clog ≥ -0.223 to clog ≥ -0.095.